Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Analytic Combinatorics of the Mabinogion and OK-Corral Urns Philippe Flajolet Based on joint work with Thierry Huillet and Vincent Puyhaubert ALEA’08 , Maresias, Brazil, April 2008 1 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Mabinogion MABINOGION Population of N sheep that bleat either A [ aah ] or B [ eeh ]. At times t = 0 , 1 , 2 , . . . , a randomly chosen sheep bleats and convinces one sheep of the other kind to change its opinion. Time to reach unanimity? Probability that one minority group wins? E.g.: French election campaign (2007): N = 60 , 000 , 000. Probability of reversing of majority of 51%? In the “fair” case ( N / 2 , N / 2), time to reach unanimity? 2 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral OK Corral OK CORRAL Population of N gangsters of gang either A or B . At times t = 0 , 1 , 2 , . . . , a randomly chosen gangster kills a member of the other group. Time to win? Probability that one minority group survives? E.g.: The OK Corral fight at Tombstone. (Wyatt Earp and Doc Holliday) 3 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Urn models (1) An urn contains balls of 2 possible colours A fixed set of rules governs the urn evolution: Place � �� � Red (A) Blue (B) Red (A) Draw : α β Blue (B) γ δ Balanced urns: α + β = γ + δ =: σ Classically: β, γ ≥ 0 and σ > 0. Convention: The ball “drawn” is not withdrawn (not taken out)! 4 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Urn models (2) All (classical) balanced 2 × 2 models are “integrable”! � � α β = An urn ; γ δ = A partial differential operator D = x α +1 y β ∂ x + x γ y δ +1 ∂ y ; = An ordinary nonlinear system { ˙ ˙ X = X α +1 Y β , Y = X γ Y δ +1 } . Refs : [Fl-Ga-Pe’05]; [Fl-Dumas-Puyhaubert’06] [Conrad-Fl’06] [Hwang-Kuba-Panholzer’07+]; [Mahmoud ⋆ ]. Cf also: Janson ⋆ ♥♥♥ 5 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral § 1. Ehrenfest & Mabinogion 6 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Ehrenfest � � − 1 1 Ehrenfest’s two chambers E = ; 1 − 1 { ˙ ˙ Formally: D = x ∂ y + y ∂ x ; X = Y ; Y = Y } ; Combinatorics of set partitions: histories from ( N , 0) to ( k , N − k ) are partitions with N − k even classes and k odd classes: � N � P [( N , 0) → ( k , N − k ) , N steps] = n ! · [ z n ] sinh k ( z ) cosh N − k ( z ) N n · k 0 1 2 3 Also: path in a special graph k / N ( N − k ) / N Also: special walks on the interval k − 1 ← − k − → k + 1 7 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Mabinogion (1) k / N ( N − k ) / N Ehrenfest: k − 1 ← − k − → k + 1 ( N − k ) / N k / N Mabinogion: k − 1 ← − k − → k + 1 + absorption Time-reversal relates M [ N ] and E [ N + 2], with fudge factors Theorem M1. Absorption time T of the Mabinogion urn: � N − 2 � P ( T = n + 1) = N − 1 n ! [ z n ](sinh z ) k − 1 (cosh z ) N − k − 1 . N n +1 k − 1 8 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Trajectories 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 9 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Mabinogion (2) Theorem M2. Probability Ω N , k of majority reversal: k = xN is initial # of A’s, with x > 1 2 ; A’s become extinct 1 − lim N log Ω N , k = log 2 + x log x + (1 − x ) log(1 − x ) . N →∞ Proof. Laplace transform + Laplace method (peak at end-point). � ∞ e − z (sinh z ) k − 1 (cosh z ) N − k − 1 dz Ω N , k = · · · 0 � 1 (1 − y ) k − 1 (1 + y ) N − k − 1 dy = · · · � N − 2 � 0 1 2 − N +1 ∼ 2 x − 1 . k − 1 N = 60 , 000 , 000: 51% → 10 − 5 , 215 ; 50.1% → 10 − 54 . 10 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Mabinogion (3) Theorem M3. Time T till absorption , when k = xN is initial number of A’s, with x < 1 2 : � T − N τ � � t 1 e − w 2 / 2 dw . √ → √ P σ N 2 π −∞ τ ( x ) = 1 1 − 2 x ; σ ( x ) 2 = x (1 − x ) 1 (1 − 2 x ) 2 + 1 2 log 2 log(1 − 2 x ). Proof. Laplace transform � ∞ � e uT � e − z � � k − 1 � � N − k − 1 sinh uz cosh uz = · · · dz E N N 0 √ Characteristic functions u = e it / N . Laplace method (with peak inside the interval) and perturbation. 11 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Experiments Distribution of absorption time UNFAIR URN FAIR URN 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 K 1 0 1 2 3 4 Normal � 12 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Mabinogion (4) Theorem M4. FAIR URN N = ν + ν : time � T till absorption T = n ) ∼ 2 n = 1 ν Ce − t e − e − 2 t ; P ( � 2 ν log ν + t ν. �� 2 j − 1 � 2 � ν/ 2 � ( i ) Exact distribution ∝ Geom . ν j =1 ( ii ) Saddle point for [ z n ](sinh z ) N when n ≈ N log N , with suitable perturbations. � ε ℓ − 1 d � T ∞ ≡ ℓ − 1 / 2 , ε ℓ ∈ Exp(1) . ℓ ≥ 1 [Simatos-Robert-Guillemin’08] [Biane-Pitman-Yor*] 13 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral § 2. Friedman & OK-Corral 14 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Friedman � � 0 1 Adverse campaign model: F = 1 0 = A counterbalance of influences [Friedman 1949] • System: { ˙ ˙ X = XY , Y = XY } is exactly solvable Distribution is expressed in terms of Eulerian numbers: rises in perms; leaves in increasing Cayley trees; other urn models [Mahmoud*] [FlDuPu06]. � A n , k u k z n 1 − u A ( z , u ) = n ! = 1 − ue z (1 − u ) n , k � n + 1 � � ( − 1) j ( k − j ) n A n , k = j 0 ≤ j ≤ k 15 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral OK Corral 100 80 60 Two gangs of m and n gunwomen 40 At any time, one shooter shoots 20 Survival probabilities? 0 0 20 40 60 80 100 Time till one gang wins? ( m , n ) = (50 , 50) 100 [Williams & McIlroy 1998] [Kingman 1999] 80 [Kingman & Volkov 2003] 60 Calamity Jane � Morris & Goscinny c 40 20 0 0 20 40 60 80 100 m + n = 100 16 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral OK Corral (1) � � � � 0 1 0 − 1 Friedman F = ; OK Corral O = ; i.e., O = −F 1 0 − 1 0 s Time reversal: P O ( m , n ց s ) = m + nP F ( s , 0 → m , n ) Theorem O1. Probability of s survivors of type A, from ( m , n ) � k − 1 �� m + n � m � s ! k m + n − s ( − 1) m − k P O ( m , n , s ) = ( m + n )! s − 1 n + k k =1 Involves generalized Eulerian numbers and expansions + identities. Theorem O2. The probability that first group survives is � m + n � m n � � 1 1 ( − 1) m − k k m + n P O S ( m , n ) = A m + n − k = ( m + n )! ( m + n )! n + k k =1 k =1 17 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral OK Corral (2) m − n Theorem O3. NEARLY FAIR FIGHTS: If √ m + n = θ , then √ � θ � 1 � 3 1 e − t 2 / 2 dt + O √ P O S ( m , n ) = . n 2 π −∞ Theorem O4. UNFAIR FIGHTS: Probability of survival with m = α n and α < 1 is exponentially small . It is related to the Large Deviation rate for Eulerian statistics. Make use of explicit GFs and usual Large Deviation techniques [Quasi-Powers, shifting of the mean] 18 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Theorem O5. [Kingman] Number of survivors: √ If ( m − n ) ≫ √ m + n then in probability S → m 2 − n 2 ; If ( m − n ) ≪ √ m + n then � λ 2 √ 3 / 8 1 e − t 2 / 2 dt . P ( S ≤ λ n 3 / 4 ) → √ 2 π −∞ Only dominant asymptotics, indirectly from Kingman et al. Get: Theorem O6. MOMENTS � m + n � m � 1 k m + n − 1 [ f ℓ ( k ) Q ( k ) + g ℓ ( k )] , E [ S ℓ ] = ( − 1) m − k ( m + n )! n − k k =1 k + k ( k − 1) where f ℓ , g ℓ are polynomials and Q ( k ) = k + · · · is Ramanujan’s k 2 = birthday paradox function. � Complex asymptotics ` a la Lindel¨ of–Rice. 19 / 20
Urn models Ehrenfest and Mabinogion Friedman and OK-Corral Conclusions? Analytic combinatorics has something to say about balanced urn models, including some nonstandard ones � = probabilistic approaches [Mahmoud–Smythe–Janson]. Analysis of imbalanced models??? Higher dimensions? “Rather surprisingly, relatively sizable classes of nonlinear systems are found to have an extra property, integrability, which changes the picture completely. Integrable systems [...] form an archipelago of solvable models in a sea of unknown, and can be used as stepping stones to investigate properties of ‘nearby’ non-integrable systems.” Eilbeck, Mikhailov, Santini, and Zakharov (2001) 20 / 20
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